Biomedical Engineering Reference
In-Depth Information
Substituting (
3.271
)in(
3.259
) leads to the coupled form (which could have
been obtained alternatively by using (
3.99
) and substituting (
3.191
)in(
3.208
) and
differentiation with respect to k
i
analogue to (
3.253
)
1
)
"
!
#
s
¼
2
X
X
3
X
3
N
3
l
k
a
k
k
a
k
i
1
þ
k
k
a
k
j
D
k
JJ
ð Þ
2k
1
n
i
n
i
i
¼
1
k
¼
1
j
¼
1
"
!
#
ð
3
:
272
Þ
2
X
3
X
N
X
3
l
k
k
a
k
i
1
3
þ
k
a
k
J
a
3
k
a
k
j
D
k
JJ
ð Þ
2k
1
n
i
n
i
:
i
¼
1
k
¼
1
j
¼
1
Regarding (
3.272
), the initial shear and bulk modulus l
0
and K
0
as well as the
relations between P
OISSON
'
S
ratio m and l
0
and K
0
and D
1
are given by (Böl and
Reese 2008)
l
0
¼
:
X
N
m
¼
3K
0
=
l
0
2
6K
0
=
l
0
þ
2
D
1
¼
l
0
1
2m
1
þ
m
:
K
0
¼
2D
1
;
l
i
;
;
ð
3
:
273
Þ
1
i
¼
1
Highly Compressible Materials. With regard to (
3.99
), (
3.209
) and (
3.253
)
1
,
the following final constitutive equation for the K
IRCHHOFF
stress tensor for highly
compressible hyperelastic materials in terms of the principal stretch is obtained
(Silber and Steinwender 2005)
n
i
n
i
:
s
¼
2
X
X
3
N
l
k
a
k
k
a
k
i
J
a
k
b
k
ð
3
:
274
Þ
i
¼
1
k
¼
1
Regarding (
3.274
), the initial shear and bulk modulus l
0
and K
0
as well as the
relations between P
OISSON
'
S
ratio m and the parameters b
i
denote
l
0
:
¼
X
N
K
0
:
¼
P
i
¼
1
2
ð
3
þ
b
i
Þ
l
i
;
m
i
¼
b
i
1
þ
2b
i
b
i
¼
m
i
;
l
i
;
: ð
3
:
275
Þ
1
2m
i
i
¼
1
For the particular case b
i
= b = const (i = 1, 2,…., N), m is the classical
P
OISSON
'
S
ratio.
Anisotropic
Materials
-
Polynomial
Representation.
With
help
of
the
G
ÂTEAUX
variation applied on (
3.216
)
g
e
¼
0
¼
!
T
Z
;
dw
G
¼
dw G
þ
eZ
;
K
1
;
K
2
; ::::
K
N
f
½
ð
Þ
=
de
½
ow G
;
K
1
;
K
2
; ::::
K
N
ð
Þ=
oG
the second P
IOLA
-K
IRCHHOFF
stress tensor of tensor-linear materials with arbitrary
anisotropic properties can be obtained (Silber 1988):
P
II
¼
o
w G
;
K
1
;
K
2
; ::::
K
N
ð
Þ
ð
4
Þ
¼
C
G
ð
3
:
276
Þ
oG
